By Oswald Veblen

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26 Exercise. , S’ = {(z,y) E R2 : x2 +y2 = 1)). Then F2(S1) is a Moebius band; that is, Fz(S’) is homeomorphic to the quotient space obtained from [0, l] x [0, l] by identifying the point (0,~) with the point (1,l - y) for each 9 E [0, 11. Remark. Some of the results in the exercises above are in [6]. If you read [6], you should be aware of two errors: (c) on p. 3 on p. 162 (cf. 5 of [7, pp. 40-411 and [8 or 91, respectively). You should also be careful to remember the standing assumption on p.

Polon. Sci. Cl. R. 19 (1958), 668. 3. 4. F. , Berlin, 1927. K. Kuratowski, Topology, Vol. I, Acad. , 1966. 5. 6. K. Kuratowski, Topology, Vol. II, Acad. , 1968. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. Sot. 71 (1951), 152-182. REFERENCES 7. 8. 9. 29 Sam B. , Vol. , 1978. E. Smithson, First countable hyperspaces, Proc. Amer. Math. Sot. 56 (1976), 325-328. Daniel E. Wulbert, Subsets of first countable spaces, Proc. Amer. Math. Sot. 19 (1968), 1273-1277. This Page Intentionally Left Blank II.

If Y is an infinite, discrete space, then the Vietoris topology for CL(Y) does not have a countable base. Let ,0 be a base for the Vietoris topology for CL(Y). Note that Proof. each A E CL(Y) is open in Y. A C (A). Thus, it follows easily that (1) A = USA for each A E CL(Y). Now, it follows immediately from (1) that if A, A’ E CL(Y) such that A # A’, then DA # a,&. jA is a one-toone function from CL(Y) into p. Hence, A RESULT ABOUT METRIZABILITY c-4 ICW)l OF CL(X) 13 I IPI- Since Y is a discrete space, CL(Y) = {A c Y : A # 0}; thus, since Y is an infinite set, CL(Y) is uncountable.

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Invariants of Quadratic Differential Forms by Oswald Veblen


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